The shape of a classic piece of diamond rough is called an “octahedron.” Imagine two 4-sided pyramids stuck together at their bases. Typically, that piece of rough is sawed into two pieces which are each then made into polished diamonds. Since diamonds jump in price exponentially with increased weight (well,not exactly, but see here for an explanation of diamond pricing), it makes more economic sense to saw that octahedron not down the middle, but just off center. Then the center of that piece of rough becomes the girdle of the larger polished diamond. (see Figure 2)
Don’t worry, there’s a reason why I’m boring you with all these details. Take a look at Figure 2. Try to imagine what the rough looks like after it’s been sawed through the plane represented by the black line. What diamond cut (shape) does it look like? If you guessed Princess cut, you’d be correct. A princess cut is really just a slightly faceted sawed diamond crystal. It was invented not for its beauty, but in order to minimize loss of diamond material. A Princess cut can easily have a yield percentage of rough of 80-90% Compare this with a round diamond (represented by the purple diamond in Figure 3) that will at best have a yield percentage of rough of about 40%.
For diamond crystals with less than perfectly formed corners, the cutter might choose a radiant cut or asscher cut. For diamond crystals that are slightly more rectangular, the diamond cutter will choose a rectangular shape, such as an Emerald cut.
Some naturally occurring diamond crystals look like flat triangles. That’s how the Trillion, Heart shape, and Pear Shape diamonds were born.
Anyway, you get the idea. With the exception of the Round Brilliant Cut, all the others were created to maximize polishing yield in different naturally occurring crystal formations.
In terms of popularity, Round Brilliant Cuts are by far the most popular shape. After rounds, princess cuts are the most popular. After princess, the numbers drop pretty steeply. I found this interesting chart at Jogia Diamond’s blog. They accumulated this data through their diamond search feature on their website. The proportions are for diamonds searched, not diamonds sold.